Signal Representation and Notation

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Introduction to Digital Filters with Audio Applications

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An exponential is any function of the form Ae^-t/τ, where t is the independent variable (often denoting time in seconds), A is an arbitrary scale factor, τ is called the time constant, and e is Euler's number (2.718281828459...). Many systems in nature exhibit exponential decay over time, because an exponential decay is produced whenever the rate of decay is proportional to how much is left. — Click for https://ccrma.stanford.edu/~jos/mdft/Exponentials.html

A sinusoid is any function of the form A sin(ω t+φ), where t is the independent variable, and A, ω, φ are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively. Sinusoidal motion is produced by any 'pure' vibration, such as that of an ideal tuning fork or mass-spring system. — Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

Spectrum analysis of sound is analogous to decomposing white light into its component colors by means of a prism — Click for https://ccrma.stanford.edu/~jos/mdft/Example_Applications_DFT.html

The subject of Digital Signal Processing (DSP) is concerned with the computational processing of digital signals. An example digital signal is music from a compact disk (CD). An example DSP task is modifying the tone of a digital audio stream, such as boosting its bass or treble. Another DSP example is compressing digital audio into MP3 format, or decoding an MP3 stream. — Click for https://ccrma.stanford.edu/~jos/filters/Book_Series_Overview.html

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Background Fundamentals

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Signal Representation and Notation

Below is a summary of various notational conventions used in digital signal processing for representing signals and spectra. For a more detailed presentation, see the elementary introduction to signal representation, sinusoids, and exponentials in [84].^A.1

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Signal Representation and Notation

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition) Copyright © 2024-09-03 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University